Related papers: Complexity of a Root Clustering Algorithm
Let $F(z)$ be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural $\varepsilon$-clusters of roots of $F(z)$ in some box region $B_0$ in the complex plane. This may be viewed as an…
In our quest for the design, the analysis and the implementation of a subdivision algorithm for finding the complex roots of univariate polynomials given by oracles for their evaluation, we present sub-algorithms allowing substantial…
We describe a subdivision algorithm for isolating the complex roots of a polynomial $F\in\mathbb{C}[x]$. Given an oracle that provides approximations of each of the coefficients of $F$ to any absolute error bound and given an arbitrary…
The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial $p$ of degree $d$ with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for…
We describe Ccluster, a software for computing natural $\epsilon$-clusters of complex roots in a given box of the complex plane. This algorithm from Becker et al.~(2016) is near-optimal when applied to the benchmark problem of isolating all…
Motivated by applications in social and biological network analysis, we introduce a new form of agnostic clustering termed~\emph{motif correlation clustering}, which aims to minimize the cost of clustering errors associated with both edges…
We describe a subroutine that improves the running time of any subdivision algorithm for real root isolation. The subroutine first detects clusters of roots using a result of Ostrowski, and then uses Newton iteration to converge to them.…
We consider the problem of approximating all real roots of a square-free polynomial $f$. Given isolating intervals, our algorithm refines each of them to a width of $2^{-L}$ or less, that is, each of the roots is approximated to $L$ bits…
We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only…
Continuous amortization is a technique for computing the complexity of algorithms, and it was first presented by the author in Burr, Krahmer, & Yap (2009). Continuous amortization can result in simpler and more straight-forward complexity…
The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in…
Let f be a univariate polynomial with real coefficients, f in R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this…
Partitioning and grouping of similar objects plays a fundamental role in image segmentation and in clustering problems. In such problems a typical goal is to group together similar objects, or pixels in the case of image processing. At the…
An effective technique for solving optimization problems over massive data sets is to partition the data into smaller pieces, solve the problem on each piece and compute a representative solution from it, and finally obtain a solution…
Tensor factorizations are computationally hard problems, and in particular, are often significantly harder than their matrix counterparts. In case of Boolean tensor factorizations -- where the input tensor and all the factors are required…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a real coefficient polynomial. They can be approximated at a low computational cost if the…
The objective of clustering is to discover natural groups in datasets and to identify geometrical structures which might reside there, without assuming any prior knowledge on the characteristics of the data. The problem can be seen as…
When approaching a clustering problem, choosing the right clustering algorithm and parameters is essential, as each clustering algorithm is proficient at finding clusters of a particular nature. Due to the unsupervised nature of clustering…